Electric-field-driven polymer entry into asymmetric nanoscale channels

The electric-field-driven entry process of flexible charged polymers such as single-stranded DNA (ssDNA) into asymmetric nanoscale channels such as the $\alpha$-hemolysin protein channel is studied theoretically and using molecular dynamics simulations. Dependence of the height of the free-energy barrier on the polymer length, the strength of the applied electric field, and the channel entrance geometry is investigated. It is shown that the squeezing effect of the driving field on the polymer and the lateral confinement of the polymer before its entry to the channel crucially affect the barrier height and its dependence on the system parameters. The attempt frequency of the polymer for passing the channel is also discussed. Our theoretical and simulation results support each other and describe related data sets of polymer translocation experiments through the $\alpha$-hemolysin protein channel reasonably well.

Figure 1

Schematic of an asymmetric channel consisting of two coaxial cylinders of different diameters and a polymer which is released from equilibrium in front of the channel (upper panel). Schematic of the free-energy landscape of the polymer when it enters from the cis side of the channel without and with applied driving electric field (lower panel).

Figure 2

Polymer entry time into a cylindrical channel in a wall versus the channel diameter. In the first two points of the plot, polymer enters from one end. In the other points, entry in folded state is also possible.

Figure 3

(a) Entry time, $\tau$, the time it takes the polymer ends to find the channel, ${\tau }_0$, and ${\tau }_0\exp \left(\frac{\Delta U}{k_{B}T}\right)$ are shown respectively with the solid, dotted, and dashed lines. Inset: Free-energy barrier versus the inverse of the electric field. (b) Entry times into a channel of the same dimensions as $\alpha$-hemolysin. Order of magnitude of the entry times and ratio entry times from cis and trans sides are in agreement with experiment [18]. Inset: Semilog plot of the histogram of the entry time, $\tau$, corresponding to data points of panel (a) for the trans side. Solid lines are guides to eye showing that semilog plot of the histograms is almost linear.

Figure 4

Schematic of a polymer under the electric field of strength $E_o$ behind a channel of diameter $d$ in a wall. (a) The polymer can be viewed as a 2D chain of blobs of size $L_{\mathrm{I}}$. (b) A polymer which is also confined laterally by a cylinder of diameter $D$. The polymer can be considered as a chain of blobs closely packed inside a cylinder of height $L_{\mathrm{II}}$. (c) A segment of the polymer entered to the channel inside which the electric field is of strength $E_i$.

Figure 5

(a) Exponents $\beta$ and $\gamma$ in relations $L\propto N^{\beta }$ and $R_{\parallel }{L}^{1/4}\propto N^{\gamma }$ for three different values of the field strengths. (b) Exponent $\alpha$ in $L\propto E^{-\alpha }$ for three different polymer lengths.

Figure 6

(a), (b) Symbols are simulation results for free-energy barrier versus the electric field for different values of the polymer length in cases I and II. In (b) the axes are scaled such that all the data sets fall on a master curve. The solid line is the fitted function of the theory, Eq. (6), to the simulation data. Insets show free-energy barrier versus $E^{-1}$. Deviations from linear behavior can be seen over large intervals of the electric field, specially in case II.

Figure 7

Distribution of $z$ component for one of end monomers of the polymer for cases I and II. The values of $z$ components corresponding to the maximum of the distributions are $z_{m,\mathrm{I}}=1.55$ and $z_{m,\mathrm{II}}=1.15$. Inset: ${\tau }_0$ versus $\Omega =L{D}^2$. Although the values of $\Omega$ for cases I and II are close to each other, the values of ${\tau }_0$ for the two cases are very different. This may result from the failure of the mean-field assumption for the distribution of the monomers in our theory.

Figure 8

Logarithm of capture time versus $\Delta V$ extracted from referenced articles (symbols) and fitted equation of our theory, Eq. (6) (solid lines). All the data sets are from polymer translocation experiments through the $\alpha$-hemolysin channel. As it can be seen, all data sets are followed well by the suggested function. Fit parameters are shown in Table .